Integrand size = 15, antiderivative size = 104 \[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a c}+\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )}{2 a c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )}{2 a c}+\frac {3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+a x}\right )}{4 a c} \] Output:
-arctanh(a*x)^3*ln(2/(a*x+1))/a/c+3/2*arctanh(a*x)^2*polylog(2,1-2/(a*x+1) )/a/c+3/2*arctanh(a*x)*polylog(3,1-2/(a*x+1))/a/c+3/4*polylog(4,1-2/(a*x+1 ))/a/c
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {-4 \text {arctanh}(a x)^3 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+6 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+6 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )}{4 a c} \] Input:
Integrate[ArcTanh[a*x]^3/(c + a*c*x),x]
Output:
(-4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]^2*PolyLog [2, -E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*PolyLog[3, -E^(-2*ArcTanh[a*x]) ] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/(4*a*c)
Time = 0.62 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6470, 6618, 6622, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{a c x+c} \, dx\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx}{c}-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a c}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {3 \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )}{c}-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a c}\) |
| \(\Big \downarrow \) 6622 |
| \(\displaystyle \frac {3 \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a}\right )}{c}-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {3 \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{a x+1}\right )}{4 a}\right )}{c}-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a c}\) |
Input:
Int[ArcTanh[a*x]^3/(c + a*c*x),x]
Output:
-((ArcTanh[a*x]^3*Log[2/(1 + a*x)])/(a*c)) + (3*((ArcTanh[a*x]^2*PolyLog[2 , 1 - 2/(1 + a*x)])/(2*a) + (ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 + a*x)])/(2* a) + PolyLog[4, 1 - 2/(1 + a*x)]/(4*a)))/c
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[k + 1, u]/ (2*c*d)), x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] & & EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.08 (sec) , antiderivative size = 593, normalized size of antiderivative = 5.70
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{c}-\frac {3 \left (\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{6}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6}\right )}{c}}{a}\) | \(593\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{c}-\frac {3 \left (\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{6}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6}\right )}{c}}{a}\) | \(593\) |
| parts | \(\frac {\ln \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )^{3}}{a c}-\frac {3 \left (\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{6 a}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a}-\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a}+\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6 a}\right )}{c}\) | \(610\) |
Input:
int(arctanh(a*x)^3/(a*c*x+c),x,method=_RETURNVERBOSE)
Output:
1/a*(1/c*arctanh(a*x)^3*ln(a*x+1)-3/c*(2/3*arctanh(a*x)^3*ln((a*x+1)/(-a^2 *x^2+1)^(1/2))-1/6*arctanh(a*x)^4+1/2*arctanh(a*x)^2*polylog(2,-(a*x+1)^2/ (-a^2*x^2+1))-1/2*arctanh(a*x)*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+1/4*poly log(4,-(a*x+1)^2/(-a^2*x^2+1))+1/6*(-I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1 ))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/ (a^2*x^2-1)+1))+I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*csgn(I*(a*x+1)^2/( a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^ (1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))+2*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^ (1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^ 3-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1 )^2/(a^2*x^2-1)+1))^2+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x ^2-1)+1))^3+2*ln(2))*arctanh(a*x)^3))
\[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \] Input:
integrate(arctanh(a*x)^3/(a*c*x+c),x, algorithm="fricas")
Output:
integral(arctanh(a*x)^3/(a*c*x + c), x)
\[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a x + 1}\, dx}{c} \] Input:
integrate(atanh(a*x)**3/(a*c*x+c),x)
Output:
Integral(atanh(a*x)**3/(a*x + 1), x)/c
\[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \] Input:
integrate(arctanh(a*x)^3/(a*c*x+c),x, algorithm="maxima")
Output:
-1/8*log(a*x + 1)*log(-a*x + 1)^3/(a*c) + 1/8*integrate((6*a*x*log(a*x + 1 )*log(-a*x + 1)^2 + (a*x - 1)*log(a*x + 1)^3 - 3*(a*x - 1)*log(a*x + 1)^2* log(-a*x + 1))/(a^2*c*x^2 - c), x)
\[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \] Input:
integrate(arctanh(a*x)^3/(a*c*x+c),x, algorithm="giac")
Output:
integrate(arctanh(a*x)^3/(a*c*x + c), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{c+a\,c\,x} \,d x \] Input:
int(atanh(a*x)^3/(c + a*c*x),x)
Output:
int(atanh(a*x)^3/(c + a*c*x), x)
\[ \int \frac {\text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {\int \frac {\mathit {atanh} \left (a x \right )^{3}}{a x +1}d x}{c} \] Input:
int(atanh(a*x)^3/(a*c*x+c),x)
Output:
int(atanh(a*x)**3/(a*x + 1),x)/c